The circular disk of 150-mm radius has a mass of 30 kg with centroidal radius of

Question

The circular disk of 150-mm radius has a mass of 30 kg with centroidal radius of gyration = 125 mm and has a concentric circular groove of 55-mm radius cut into it. A steady force T is applied at an angle ? to a cord wrapped around the groove as shown. If T = 28 N, ? = 0, ?s = 0.10, and ?k = 0.08, determine the angular acceleration ? of the disk, the acceleration a of its mass center G, and the friction force F which the surface exerts on the disk. The angular acceleration ? is positive if counterclockwise, negative if clockwise; the acceleration a is positive if to the right, negative if to the left; and the friction force F is positive if to the right, negative if to the left.

Explanation / Answer

k = 125 mm = 0.125 m r = 0.15 m M = 30 kg I = M*k^2 = 0.469 kg-m^2 let friction is F. for the cord to not slip on the groove, alpha = a/0.055      --->(1) Now, net torque = I*alpha => T*0.055 - F*0.15 = 0.469 *(a/0.055) => 28*0.055 - (uk*Mg)*0.15 = 8.527*a => 1.54 - (0.08*30*9.8)*0.15 = 8.527*a => a = - 0.233 m/s^2 alpha = 0.233/0.055 = 4.236 rad/s^2 F = - 0.08*30*9.8 = - 23.52 N

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